Soliton resolution for the Hirota equation with weighted Sobolev initial data

In this work, the $overline{partial}$ steepest descent method is employed to investigate the soliton resolution for the Hirota equation with the initial value belong to weighted Sobolev space $H^{1,1}(mathbb{R})={fin L^{2}(mathbb{R}): f,xfin L^{2}(mathbb{R})}$. The long-time asymptotic behavior of the solution $q(x,t)$ is derived in any fixed space-time cone $C(x_{1},x_{2},v_{1},v_{2})=left{(x,t)in mathbb{R}timesmathbb{R}: x=x_{0}+vt ~text{with}~ x_{0}in[x_{1},x_{2}]right}$. We show that solution resolution conjecture of the Hirota equation is characterized by the leading order term $mathcal {O}(t^{-1/2})$ in the continuous spectrum, $mathcal {N}(mathcal {I})$ soliton solutions in the discrete spectrum and error order $mathcal {O}(t^{-3/4})$ from the $overline{partial}$ equation.